Over the last 2 decades Lavra became a pilgrimage site for Orthodox Christians with many saints' bodies on display on the caves. I was shocked by the crowd level - basically there was almost no gap. The signs were often poorly lit and with people pushing you had little time to read them. Almost no signs, especially explaining who lays there, were in English.
That being said, the place is historically amazing. The above ground churches have been restored and maintained very well. If you get a chance to do a tour with a guide, it is worth it for the stories and the history. Some days the entry to grounds is free due to religious holidays. Caves can always be visited for free, just go to the side entrance (200 meters to the right of main entrance). They will even tell you so at the ticket booth, if you speak Russian.
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC.